# When are we justified in making assumptions in questions

1. The density at $$20^\circ\mathrm{C}$$ of a $$\pu{0.500 M}$$ solution of acetic acid in water is $$\pu{1.0042 g/mL}$$. What is the molality of the solution? The molar mass of acetic acid is $$M(\ce{CH3CO2H})=\pu{60.05 g/mol}$$.
2. The density at $$20^\circ\mathrm{C}$$ of a $$\pu{0.258 M}$$ solution of glucose in water is $$\pu{1.0173 g/mL}$$, and the molar mass of glucose is $$M(\ce{Glc}) = \pu{180.2 g/mol}$$. What is the molarity of the solution?

In question one we can assume a 1 liter solution and in question two we can assume a 100 gram sample. I understand these assumptions are needed to find other information, but when are we justified in making these assumptions?

As it happens, there's no need to assume any particular quantities to solve this problem; instead, you can use algebraic substitutions using the known values given and common definitions to solve for either molarity or molality directly. Let $M = \frac{m_{solute}}{n_{solute}}$ be the molar mass of the solute, $\rho = \frac{m_{solution}}{V_{solution}}$ be the mass density of the solution, $c = \frac{n_{solute}}{V_{solution}}$ be the molarity, and $b = \frac{n_{solute}}{m_{solvent}}$ be the molality, and note that $m_{solvent} = m_{solution} - m_{solute}$. Then, we have:
\begin{align} b\ & = \frac{n_{solute}}{m_{solution} - m_{solute}}\\ & = \frac{n_{solute}}{\rho \cdot V_{solution} - M \cdot n_{solute}}\\ & = \frac{n_{solute}}{\rho \cdot \frac{n_{solute}}{c} - M \cdot n_{solute}}\\ & = \frac{c}{\rho - M \cdot c}\\ \end{align}
You can rearrange this equation to solve for $c$ (molarity). Note, of course, that units need to match, but it's trivial to convert between grams and kilograms, mililiters and liters, etc.